Galaxy rotation curves

Why do the outermost stars of a galaxy orbit as fast as the inner ones when the outer planets do not?

Earth goes round the Sun at about 29.8 kilometres a second; Neptune, thirty times farther out, manages about 5.4. Nobody finds that odd — it is what gravity is supposed to do. Now point a radio telescope at a spiral galaxy and measure how fast material orbits at each radius. The speed climbs, levels off, and then simply refuses to fall. Out where the starlight has faded to nearly nothing, gas is still circling at close to the speed of material well inside it.

So either gravity behaves differently on that scale, or there is a great deal of matter out there that emits no light. The interesting question is not which answer sounds better. It is: what could possibly decide between them?

Start by asking why Neptune should be slow. For a circular orbit, the inward pull must supply the centripetal acceleration, and that gives orbital speed as the square root of GM over r. Take Earth's 29.8 km/s, divide by the square root of Neptune's 30.07 astronomical units, and you get 5.43 — which is Neptune's measured speed. The rule works.

But notice the quiet assumption inside it. That M is the mass enclosed by the orbit, and in the solar system it barely changes: the Sun is about 99.86 per cent of the total, so every planet orbits essentially the same M. Speed falls as one over the square root of radius only because the mass has already run out.

A galaxy is not built that way. Its mass is spread through the disk. Newton's shell theorem tells us that for a roughly spherical distribution, only the mass inside your orbit pulls you inward on net — the shells outside cancel. So the same formula applies, but now with M(r), a quantity that grows as you move out.

Turn the crank the other way. If the observed speed is flat, then GM(r)/r is constant, which means M(r) rises in direct proportion to r. Each additional kiloparsec of radius must add the same additional mass, indefinitely, well past the point where the galaxy looks empty.

Does the light do that? Emphatically not. Disk brightness falls off roughly exponentially, so beyond a few scale lengths, going further out adds essentially no luminosity. The mass keeps accumulating; the light stops. That mismatch is the whole observation, and the decisive data came from radio rather than optical astronomy — the 21-centimetre line of neutral hydrogen traces gas far outside the visible disk, and it was that gas, mapped in the 1970s alongside Vera Rubin and Kent Ford's optical work, that showed the curve staying flat rather than merely failing to fall quickly.

Now the hard part. An equation with the wrong answer can be repaired at either end. Add unseen mass on the right, or change the law on the left. Modified Newtonian dynamics, proposed by Milgrom in 1983, alters the force below an acceleration of roughly 1.2 x 10^-10 metres per second squared and reproduces spiral rotation curves with almost embarrassing accuracy — often from the visible matter alone, with one free parameter per galaxy. And it predicts tight regularities that are genuinely awkward for the mass hypothesis: why should invisible matter, distributed by its own dynamics, track the visible matter closely enough to make the baryonic Tully-Fisher relation as narrow as it is?

So rotation curves, by themselves, do not settle this. What settles it is that the question is asked in other places too. The relative heights of the acoustic peaks in the cosmic microwave background require matter that does not interact with light. Big-bang nucleosynthesis caps ordinary matter at about five per cent of the critical density while total matter comes out near thirty. Gravitational lensing weighs clusters independently of any orbit, and in the Bullet Cluster the lensing mass stayed with the galaxies while the hot gas — most of the ordinary matter — was stripped and left behind. Modified gravity fits galaxies and struggles with clusters; extra mass fits clusters and cosmology and struggles to explain why galaxies are so regular.

The honest position is that one hypothesis wins on breadth, not on any single measurement — and that it still has an outstanding debt: decades of increasingly sensitive direct-detection experiments have not caught a dark matter particle.

Two refinements matter. First, "flat" is a statement about the outer curve; the inner rise is set by ordinary matter, and nobody disputes that part. Second — and this is the step most often skipped — the missing matter cannot simply be ordinary matter that happens to be dark. Faint stars, planets and black holes are all made of baryons, and the abundance of deuterium left over from the first minutes of the universe limits the total baryon budget independently of any telescope; microlensing surveys have also failed to find enough compact dark objects in the halo.

It is worth naming what the evidence does not do. It does not identify a particle, or tell us its mass, or rule out that gravity is modified in some deeper theory that also reproduces the cosmological data. Calling the answer "dark matter" names a gap in the accounts, not a discovery.

• How the acoustic peaks of the cosmic microwave background weigh ordinary and non-ordinary matter separately.
• Why the Bullet Cluster's separation of lensing mass from hot gas is hard for any modified-gravity account.

Every term the collection defines is gathered in the glossary.